Dimension Mode is a game mode that has double fruits, half normal, and other half that's in the other "dimension", the latter you can't interact with until you eat the regular fruit, and vice versa. When you eat a fruit, your non-transparent (in current dimension) body becomes transparent, and vice versa - because the dimensions are switched. If you keep going without eating food, eventually the entire body becomes visible and moves to the current dimension, and the next fruit you eat make it all transparent (since it stays that dimension), allowing you to switch up your coil completely and also switch directions late game.
Depending on what game modes you have active (via Blender Mode) other things like poison fruit, flags, stone, etc, will also be in the other dimension. Winged Fruit will still be able to interact with each other and objects despite which dimension the player is in.